Make the vinaigrette: Whisk together vinegar, mustard, and reserved cooking water in a small bowl. Gradually add oil, whisking until emulsified. Transfer potatoes to a medium bowl, and drizzle with vinaigrette. Add onion, cornichons, capers, parsley, remaining 1 1/2 teaspoons salt, and the pepper. Using a large spatula, gently stir to combine.